MR measurements of water diffusion in organs and tissues having an orderly, oriented structure, such as skeletal, cardiac, and uterine muscle, portions of the kidney, the lens, and white matter, exhibit diffusion anisotropy (i.e., a dependence of the diffusively on direction). The development of quantitative MRI measures of diffusion anisotropy is deemed to have important biological and clinical applications, helping clinicians infer microstructural characteristics of normal tissues that are undetected using other techniques, as well as pathological changes in tissue microstructure. This microstructural information may be useful in arriving at a correct diagnosis, as well as choosing and implementing appropriate therapies.
However, while a qualitative indication of diffusion anisotropy can be obtained by inspecting diffusion weighted images (DWIs), a quantitative assessment is more problematic. On theoretical grounds, it has been predicted that currently used indices of diffusion anisotropy derived from DWIs or from two or three apparent diffusion coefficients (ADCs) measured in perpendicular directions are not quantitative. In particular, they are rotationally variant because their values depend upon the direction of the applied diffusion gradients and the orientation of structures within each voxel. See, e.g., P. J. Basser, J. Mattiello, D. LeBihan. "MR diffusion tensor spectroscopy and imaging", Biophys J 66, 259-267 (1994), which is herein incorporated by reference. Further, it has been experimentally observed in living monkey brain that these indices which are calculated directly from ADCs generally underestimate the degree of diffusion anisotropy, and that this orientational artifact can be so severe as to make some highly anisotropic white matter structures appear completely isotropic, indistinguishable from gray matter. See, Pierpaoli, Carlo, and Basser, Peter J., "Toward a Quantitative Assessment of Diffusion Anisotropy", Magnetic Resonance in Medicine, vol. 36, pp. 893-906 (December 1996).
This orientational artifact can be eliminated by using rotationally invariant anisotropy indices that are functions of the eigenvalues of the diffusion tensor. These rotationally invariant anisotropy measures have values that are independent of the laboratory frame of reference, the direction of the applied diffusion gradients, and the orientation of the tissue structures within each voxel. Some of these rotationally invariant anisotropy measures require that the eigenvalues be sorted in magnitude order, while others do not require such sorting.
Nevertheless, even though these rotationally invariant indices elucidate anisotropy independent of orientational artifacts, these indices are still susceptible to noise inherent in the DWIs. Specifically, it has been demonstrated by Monte Carlo simulations that measurement noise introduces a bias in rotationally invariant diffusion anisotropy measures such that isotropic structures appear anisotropic and anisotropic structures appear more anisotropic. See, Pierpaoli, Carlo, and Basser, Peter J., "Toward a Quantitative Assessment of Diffusion Anisotropy", Magnetic Resonance in Medicine, vol. 36, pp. 893-906 (December 1996).
Further, due to the noise-induced bias in the measured eigenvalues, rotationally invariant anisotropy measures that require the eigenvalues to be sorted by their magnitude (such as the ratio of the largest and smallest eigenvalues, .lambda..sub.1 /.lambda..sub.3) are susceptible to a sorting bias that systematically overestimates of the degree of diffusion anisotropy. Moreover, once the eigenvalues (and/or eigenvectors) are sorted, the assumptions of random sampling are violated so that standard statistical tests used to analyze their distributions no longer apply.
In inhomogeneous tissues, attempts to increase SNR, either by increasing voxel size or by averaging the signal intensity of DWIs (or diffusion tensors, or rotationally invariant indices) over a region of interest, introduce a partial volume artifact that causes one to underestimate diffusion anisotropy by averaging a nonuniform distribution of fiber-tract directions. Therefore, such averaging techniques cannot be applied generally to mitigate noise and bias in the anisotropy indices.
It is further noted that the diffusion tensor may be used to construct diffusion ellipsoid images to elucidate anisotropy. The diffusion ellipsoids are surfaces of constant mean-squared displacement of diffusing water molecules at some time .tau. after they are released at the center of each voxel. The diffusion ellipsoids summarize the information contained in the diffusion tensor. The degree of diffusion anisotropy is embodied in the shape or eccentricity of the diffusion ellipsoid; the bulk mobility of the diffusing species is related to the size of the diffusion ellipsoid; and the preferred directions of diffusion are indicated by the orientation of the diffusion ellipsoid. If the tissue were isotropic, then the water mobility would be the same in all directions, and these surfaces would be spherical. However, if the medium were anisotropic, like brain white matter, then the mobility would depend upon the direction in which it is measured, and these surfaces would be ellipsoidal. Although the diffusion ellipsoids embody the information contained in the diffusion tensor, they are subject to the same artifacts that affect other rotationally invariant anisotropy measures.
Thus, it may be understood that anisotropy measures are susceptible and sensitive to background noise inherent in all DWIs. This effect influences the mean and variance of all diffusion anisotropy measures estimated from DWIs, and thus their accuracy and precision.
There remains a need, therefore, for further improvements in quantitatively assessing diffusion anisotropy, and particularly, a need for an anisotropy measure or index which is not only rotationally invariant but also is immune to noise.